integration limits in gamma pdf convolution

fisher garrry

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In the proof for proposition 3.1 they use the limits 0 to a. In (3.2) they use integration limits \(\displaystyle -\infty\) to \(\displaystyle \infty\). The gamma function is defined from 0 to \(\displaystyle \infty\). Why is then not the integration limits in the proof for proposition 3.1 0 to \(\displaystyle \infty\) as (3.2) limits are constrained to possible values for the distribution as far as I can see? And what are then the limits they use to integrate to find the value for C in the proof for proposition 3.1?


The theory is taken from Ross A first course in probability


Tap image twice to read it.
 

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I managed to enlarge your pictures enough to figure out what is going on.

They are applying equation (3.2) as stated, which integrates from -∞ to +∞; but because the function f(y) is defined only for y>0, both y and a-y in the integrand must be positive, which restricts y to the interval 0<y<a. For y outside that domain, the integrand is not defined.
 
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